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150 VIII DIFFERENTIAL CALCULUS

When E has finite dimension n and F has finite dimension m, /'(*<»)
can thus be identified to a matrix with m rows and n columns; this matrix
will be determined in Section 8.10.

Examples

(8.1.2) A constant function is differentiable at every point of A, and its
derivative is the element 0 of jSf(E; F).

(8.1.3) The derivative of a continuous linear mapping u of E into F exists
at every point xeE and Du(x) = u.

For by definition u(x₀) + u(x - #₀) = «(*)•

(8.1.4) L*f E, F, G be three Banach spaces, (x₉y)-+[x'y] a continuous
bilinear mapping of E x F into G. Then that mapping is dlfferentioble
at every point (x, y) e E x F and the derivative is the linear mapping

For we have

[(x + s)-(y+ty\-[x-y]-[x-t]-[s-y]'*ls-t]

and by assumption, there is a constant c> 0 such that \\[s • t]\\ < c * pi • ||f ||
(5.5.1). For any e > 0, the relation sup(||^||, ||f||) « ||(j» Oil < fi/c implies
therefore

- (y + 01 -[*•?]-[*•')-[*• y]ll < e life OH

which proves our assertion.

That result is easily generalized to a continuous multilinear mapping,

(8.1.5) Suppose F = F! x F₂ x ••- x F_m is a product of Banach spaces,
andf= (/!,.. .,/J a continuous mapping of an open subset A of E into F.
/« orrfer that f be differentiable atx₀,a necessary and sufficient condition is that
each f, be differentiable at x₀, and then f(x₀) » (/((jc₀)»... _f/;(jc₀» (wAw
JSP(E; F) /j identified with the product of the spaces J5f(E; F,)),

Indeed, any linear mapping u of E into F can be written in a unique
way u = (w_l9..., w_m), where «_ris a linear mapping of E into F|_f and we

have by definition \\u(x)\\ = supdlw^z)!!,..., \\u_m(x)\\), whence it follows



	150 VIII DIFFERENTIAL CALCULUS When E has finite dimension n and F has finite dimension m, /'(<») can thus be identified to a matrix* with m rows and n columns; this matrix will be determined in Section 8.10.

	Examples (8.1.2) A constant function is differentiable at every point of A, and its derivative is the element 0 of jSf(E; F). (8.1.3) The derivative of a continuous linear mapping u of E into F exists at every point xeE and Du(x) = u. For by definition u(x₀) + u(x - #₀) = «()• (8.1.4) Lf E, F, G be three Banach spaces, (x₉y)-+[x'y] a continuous bilinear mapping of E x F into G. Then that mapping is dlfferentioble at every point (x, y) e E x F and the derivative is the linear mapping

	For we have [(x + s)-(y+ty\-[x-y]-[x-t]-[s-y]'ls-t]* and by assumption, there is a constant c> 0 such that \\[s • t]\\ < c * pi • \|\|f \|\| (5.5.1). For any e > 0, the relation sup(\|\|^\|\|, \|\|f\|\|) « \|\|(j» Oil < fi/c implies therefore - (y + 01 -[•?]-[•')-[*• y]ll < e life OH

	which proves our assertion. That result is easily generalized to a continuous multilinear mapping,

	(8.1.5) Suppose F = F! x F₂ x ••- x F_m is a product of Banach spaces, andf= (/!,.. .,/J a continuous mapping of an open subset A of E into F. /« orrfer that f be differentiable atx₀,a necessary and sufficient condition is that each f, be differentiable at x₀, and then f(x₀) » (/((jc₀)»... _f/;(jc₀» (wAw JSP(E; F) /j identified with the product of the spaces J5f(E; F,)), Indeed, any linear mapping u of E into F can be written in a unique way u = (w_l9..., w_m), where «_ris a linear mapping of E into F\|_f and we have by definition \\u(x)\\ = supdlw^z)!!,..., \\u_m(x)\\), whence it follows